Fermionic Modular Categories and the 16-fold Way

TL;DR

This paper explores the structure of fermionic topological phases through super-modular categories, proposing a 16-fold way conjecture for their modular extensions and analyzing their properties and classifications.

Contribution

It formulates a 16-fold way conjecture for minimal modular extensions of super-modular categories, connecting fermionic phases with categorical gauging of fermion parity.

Findings

Proposes a categorical 16-fold way conjecture for super-modular categories.

Analyzes properties like fermions in twisted Drinfeld doubles and Verlinde formulas.

Provides explicit extensions of $PSU(2)_{4m+2}$ for classification purposes.

Abstract

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of $PSU(2)_{4m+2}$ with an eye towards a classification of the low-rank cases.